CS 111 Homework 1
Due Friday, October 12, 2:10 PM
Problem 1.
a) Give the exact formula (as a function of n) for the number of times hello is printed by
Algorithm NiHao below. First express it using summation, and then give a closedform
expression.
b) Giv
Review for Quiz 1
1
Summations
n ( n 1)
i
2
i 1
n
n 1
1
a
i
a
when a 0, a 1
1 a
i 0
n
n
e.g.,
i
n
n 1
2
1
2
4
.
2
2
1
i 0
2
Binomial expansion
n k n k
a b a b
k 0 k
In particular, if we choose a 1, b 1
n
n
n
we get 2
k 0 k
n
n
3
Bounding sums
Upper
ENGL 103/Swenck
Critical Thinking Inventory
Beyond Feelings by Vincent Ruggerio
Answering the following questions will help you take inventory of the habits
and attitudes that affect your thinking. I am not requiring that you complete
this worksheet as an
NAME
Signature
Math 111  Section 001
Final Exam
March 25, 2006
Problem
Score
1
10 
2
10 
3
15 
4
10 
5
15 
6
10 
7
10 
8
10 
9
10 
10
10 
11
10 
12
10 
13
10 
Total
140
Show all work for full credit.
Problems 3 and 5 are worth 15 points each
Union High School
Drivers Education
Name: _
Period:_
Date: _
Chapter 1
The New Jersey Driver License System
1.
Driving is a _.
2.
Motorists must always carry a valid _ in addition to a
current _ and proof of valid _,
while operating a vehicle on New Jerse
ATaleofTwoCities
BookIII,Chapter1:"InSecret"
1. How does Charles come to realize the extreme danger he's placed himself by returning to France at this
time?
2.
3.
4.
5.
What is the full significance of the chapter's title?
Of what is Charles reminded as h
Book I: "Recalled to Life"
Book I, Chapter 1: "The Period"
1. What is the chronological setting of this opening chapter? What clues enable us to determine "The Period"?
2.
How does Dickens indicate the severity of social conditions in both France and Engl
Math 131 Lab4 Name 34; 31 Sec
Feb. 710, 2012 Name Sec
Name Sec
Day and time of lab session
Lab 4 serves as practice for computing derivatives using difference quotients. Complete the following parts with the collaboration of your
lab partners (dont
Math 131 Lab2 Name (scribe) ! no Ki 8
January 2427, 2012 Name Ml} Qt [g S
Name i? ii
Day and time of lab session had
8
, 8
Q lmlolo
The goal of Lab 2 is to learn how to calculate the limits of functions using the appropriate limit rules and to be able t
Eigenvalues and Eigenvectors
> restart;
> with(LinearAlgebra):
We illustrate the process of finding eigenvalues and eigenvectors with a 2x2 matrix.
> A:=Matrix(2,2,[1,1],[4,1]);
11
A :=
41
We will need to use the 2x2 identity matirx.
> Id:=IdentityMatrix(
CHAPTER 2
Separable Equations
Definition. A rstorder dierential equation
dy
= f (x, y )
dx
is said to be separable if
f (x, y ) = g (x) h(y ),
where g depends only on x and h depends only on y .
Example.
dy x 5
1
(1)
=
= (x 5) 2 , so this equation is sep
CHAPTER 3
Direction Fields and Solution Curves
Maple. See direction elds.mw or direction elds.pdf.
Vertical motion with a velocitydependent drag force
+

The eect of gravity is
dv
=
dt
g =) (by Newtons Second Law)
dv
= mg .
dt
Now include a drag force (
Maple Worksheets for Ordinary Differential Equations
Complimentary software to accompany the textbook:
Differential Equations: Concepts, Methods, and Models (20012002 Edition)
Leigh C. Becker
Department of Mathematics
Christian Brothers University 650 Ea
Abrupt Changes and the Unit Step (Heaviside) Function
O restart;with(inttrans);
addtable, fourier, fouriercos, fouriersin, hankel, hilbert, invfourier, invhilbert, invlaplace, invmellin,
laplace, mellin, savetable
We will begin by again plotting Heaviside
Distinct Complex Eigenvalues
> restart:with(LinearAlgebra):with(plots):with(DEtools):
We wish to solve the planar homogeneous linear system
dx
= 3 xK2 y
dt
3K
2
or r' = Ar with A =
.
dy
4K
1
= 4 xKy
dt
We begin with the matrix A.
> A:=Matrix(2,2,[3,2],[4
O
Integrals, Partial Fractions, and Integration by Parts
In this worksheet, we show how to integrate using Maple, how to explicitly implement integration by
parts, and how to convert a proper or improper rational fraction to an expression with partial fra
The Logistic Population Model (Page 156 #21 and Page 157 #30)
O restart:with(plots):
Suppose the per capita rate of growth is k1 , the per capita rate of death from natural causes is k2 , and the
per interaction death rate from intraspecies competition is
Euler's Method  An Example
O restart:with(plots):with(DEtools):
From algebra we recall that there are precise methods to solve linear and quadratic equations. There are
even formulas for cubic and quartic equations. However, we cannot find an exact solut
Direction Fields and Solution Curves
We begin by reinitializing the Maple "kernel" and loading the additional packages that we are most likely
to use, the plots package and the DEtools package. The plots package contains extra plotting commands
including
FirstOrder Linear Equations and Sensitive Initial Conditions
Example 1. We look at the first order linear differential equation
dx
= 10 t K 2 tx.
dt
O restart:with(plots):with(DEtools):
We enter the differential equation.
O deq:= diff(x(t),t)=10*t2*t*x(
Distinct Real Eigenvalues
> restart:with(LinearAlgebra):with(plots):with(DEtools):
We wish to solve the planar homogeneous linear system
dx
= xCy
dt
11
or x' = Ax with A =
.
dy
41
= 4 xCy
dt
We begin with the matrix A.
> A:=Matrix(2,2,[1,1],[4,1]);
A :=
1
Functions of Two Variables
O restart:with(plots):with(DEtools):
The graph of a function z = F x, y is a surface in threedimensional space. We graph the function
x
z = F x, y = x eK 2 K y 2 .
O z:=x*exp(x^2y^2);
x
z := x eK
2 K y2
We look at a plot of t